Download Actuarial Exam Practice Problem Set 4

Math 370/408, Spring 2008
Prof. A.J. Hildebrand
Actuarial Exam Practice Problem Set 4
About this problem set: These are problems from Course 1/P actuarial exams that I
have collected over the years, grouped by topic, and ordered by difficulty. All of the exams have
appeared on the SOA website http://www.soa.org/ at some point in the past, though most of
them are no longer there. The exams are copy-righted by the SOA.
Note on the topics covered: This problem set covers problems on joint (multivariate)
distributions (Chapter 4 of Hogg/Tanis).
Note on the ordering of the problems: The problems are loosely grouped by topic, and
very roughly ordered by difficulty (though that ordering is very subjective). The last group of
problems (those with triple digit numbers) are harder, either because they are conceptually more
difficult, or simply because they require lengthy computations and may take two or three times
as long to solve than an average problem.
Answers and solutions: I will post an answer key and hints/solutions on the course
webpage, www.math.uiuc.edu/∼hildebr/370.
1
Math 370/408 Spring 2008
Actuarial Exam Practice Problem Set 4
1. [4-1]
Let X and Y be discrete random variables with joint probability function
(
2x+y
for (x, y) = (0, 1), (0, 2), (1, 2), (1, 3),
12
p(x, y) =
0
otherwise.
Determine the marginal probability function of X.


1/6 for x = 0,
(A) p(x) = 5/6 for x = 1,


0
otherwise.


1/4 for x = 0,
(B) p(x) = 3/4 for x = 1,


0
otherwise.


1/3 for x = 0,
(C) p(x) = 2/3 for x = 1,


0
otherwise.

2/9 for x = 1,



3/9 for x = 2,
(D) p(x) =
4/9 for x = 3,



0
otherwise.


for x = 0,
y/12
(E) p(x) = (2 + y)/12 for x = 1,


0
otherwise.
2. [4-2]
A car dealership sells 0, 1, or 2 luxury cars on any day. When selling a car, the dealer also
tries to persuade the customer to buy an extended warranty for the car. Let X denote
the number of luxury cars sold in a given day, and let Y denote the number of extended
warranties sold, and suppose that

1/6 for (x, y) = (0, 0),




1/12 for (x, y) = (1, 0),



1/6 for (x, y) = (1, 1),
P (X = x, Y = y) =

1/12 for (x, y) = (2, 0),





1/3 for (x, y) = (2, 1),



1/6 for (x, y) = (2, 2).
What is the variance of X?
(A) 0.47
(B) 0.58
(C) 0.83
(D) 1.42
(E) 2.58
3. [4-3]
An actuary determines that the annual numbers of tornadoes in counties P and Q are
jointly distributed as follows:
2
Math 370/408 Spring 2008
Actuarial Exam Practice Problem Set 4
Annual number of tornadoes in
county Q
0
1
2
3
Annual number
0
0.12
0.06
0.05
0.02
of tornadoes
1
0.13
0.15
0.12
0.03
in county P
2
0.05
0.15
0.10
0.02
Calculate the conditional variance of the annual number of tornadoes in county Q, given
that there are no tornadoes in county P.
(A) 0.51
(B) 0.84
(C) 0.88
(D) 0.99
(E) 1.76
4. [4-4]
A diagnostic test for the presence of a disease has two possible outcomes: 1 for disease
present and 0 for disease not present. Let X denote the disease state of a patient, and let
Y denote the outcome of the diagnostic test. The joint probability function of X and Y is
given by:

0.800 for (x, y) = (0, 0),



0.050 for (x, y) = (1, 0),
P (X = x, Y = y) =
0.025 for (x, y) = (0, 1),



0.125 for (x, y) = (1, 1).
Calculate Var(Y |X = 1).
(A) 0.13
(B) 0.15
(C) 0.20
(D) 0.51
(E) 0.71
5. [4-5]
Let X and Y be random losses with joint density function
f (x, y) = e−x−y
for x > 0 and y > 0.
An insurance policy is written to reimburse X + Y . Calculate the probability that the
reimbursement is less than 1.
(A) e−2
(B) e−1
(C) 1 − e−1
(D) 1 − 2e−1
(E) 1 − 2e−2
6. [4-6]
The waiting time for the first claim from a good driver and the waiting time for the first
claim from a bad driver are independent and follow exponential distributions with means
6 years and 3 years, respectively. What is the probability that the first claim from a good
driver will be filed within 3 years and the first claim from a bad driver will be filed within
2 years?
1
(A) 18
1 − e−2/3 − e−1/2 + e−7/6
(B)
1 −7/6
18 e
−2/3
(C) 1 − e
− e−1/2 + e−7/6
3
Math 370/408 Spring 2008
Actuarial Exam Practice Problem Set 4
(D) 1 − e−2/3 − e−1/2 + e−1/3
(E) 1 − 31 e−2/3 − 16 e−1/2 +
1 −7/6
18 e
7. [4-7]
A company has two electric generators. The time until failure for each generator follows
an exponential distribution with mean 10. The company will begin using the second
generator immediately after the first one fails. What is the variance of the total time that
the generators produce electricity?
(A) 10
(B) 20
(C) 50
(D) 100
(E) 200
8. [4-8]
A joint density function is given by
(
kx for 0 < x < 1, 0 < y < 1,
f (x, y) =
0
otherwise.
where k is a constant. What is Cov(X, Y )?
(A) −1/6
(B) 0
(C) 1/9
(D) 1/6
(E) 2/3
9. [4-9]
A device contains two components. The device fails if either component fails. The joint
density function of the lifetimes of the components, measured in hours, is f (s, t), where
0 < s < 1 and 0 < t < 1. What is the probability that the device fails during the first half
hour of operation?
Z 0.5 Z 0.5
(A)
f (s, t)dsdt
0
Z
1
0
0.5
Z
(B)
f (s, t)dsdt
0
Z
0
1
Z
1
(C)
f (s, t)dsdt
0.5 0.5
Z 0.5 Z 1
(D)
1
Z
0.5
Z
f (s, t)dsdt +
0
Z
0
0.5
Z
f (s, t)dsdt
0
1
(E)
Z
0
1
Z
f (s, t)dsdt +
0
0.5
0.5
f (s, t)dsdt
0
0
10. [4-10]
Let X and Y be continuous random variables with joint density function
(
15y for x2 ≤ y ≤ x,
f (x, y) =
0
otherwise.
Let g be the marginal density function of Y . Which of the following represents g?
(
15y for 0 < y < 1,
(A) g(y) =
0
otherwise.
4
Math 370/408 Spring 2008
(B)
(C)
(D)
(E)
Actuarial Exam Practice Problem Set 4
(
15y 2 /2 for x2 < y < x,
g(y) =
0
otherwise.
(
15y 2 /2 for 0 < y < 1,
g(y) =
0
otherwise.
(
15y 3/2 (1 − y 1/2 ) for x2 < y < x,
g(y) =
0
otherwise.
(
15y 3/2 (1 − y 1/2 ) for 0 < y < 1,
g(y) =
0
otherwise.
11. [4-11]
Let X represent the age of an insured automobile involved in an accident. Let Y represent
the length of time the owner has insured the automobile at the time of the accident. X
and Y have joint probability density function
(
1
(10 − xy 2 ) for 2 ≤ x ≤ 10 and 0 ≤ y ≤ 1,
f (x, y) = 64
0
otherwise.
Calculate the expected age of an insured automobile involved in an accident.
(A) 4.9
(B) 5.2
(C) 5.8
(D) 6.0
(E) 6.4
12. [4-12]
Let X and Y be continuous random variables with joint density function
(
24xy for 0 < x < 1 and 0 < y < 1 − x,
f (x, y) =
0
otherwise.
Calculate P (Y < X|X = 1/3).
(A) 1/27
(B) 2/27
(C) 1/4
(D) 1/3
(E) 4/9
13. [4-13]
Let X and Y be random losses with joint density function
f (x, y) = 2x for 0 < x < 1 and 0 < y < 1.
An insurance policy is written to cover the loss X + Y . The policy has a deductible of 1.
Calculate the expected payment under the policy.
(A) 1/4
(B) 1/3
(C) 1/2
(D) 7/12
(E) 5/6
14. [4-14]
Let T1 and T2 represent the lifetimes in hours of two linked components in an electronic
device. The joint density function for T1 and T2 is uniform over the region defined by
0 ≤ t1 ≤ t2 ≤ L, where L is a positive constant. Determine the expected value of the sum
of the squares of T1 and T2 .
(A) L2 /3
(B) L2 /2
(C) 2L2 /3
5
(D) 3L2 /4
(E) L2
Math 370/408 Spring 2008
Actuarial Exam Practice Problem Set 4
15. [4-15]
A company offers a basic life insurance policy to its employees, as well as a supplemental
life insurance policy. To purchase the supplemental policy, an employee must first purchase
the basic policy. Let X denote the proportion of employees who purchase the basic policy,
and Y the proportion of employees who purchase the supplemental policy. Let X and
Y have the joint density function f (x, y) = 2(x + y) on the region where the density is
positive. Given that 10% of the employees buy the basic policy, what is the probability
that fewer than 5% buy the supplemental policy?
(A) 0.010
(B) 0.013
(C) 0.108
(D) 0.417
(E) 0.500
16. [4-16]
The future lifetimes (in months) of two components of a machine have the following joint
density function:
(
6
(50 − x − y) for 0 < x < 50 − y < 50,
f (x, y) = 125,000
0
otherwise.
What is the probability that both components are still functioning 20 months from now?
Z 20 Z 20
6
(50 − x − y)dydx
(A)
125, 000 0
0
Z 30 Z 50−x
6
(B)
(50 − x − y)dydx
125, 000 20 20
Z 30 Z 50−x−y
6
(50 − x − y)dydx
(C)
125, 000 20 20
Z 50 Z 50−x
6
(D)
(50 − x − y)dydx
125, 000 20 20
Z 50 Z 50−x−y
6
(50 − x − y)dydx
(E)
125, 000 20 20
17. [4-17]
A device contains two circuits. The second circuit is a backup for the first, so the second is
used only when the first has failed. The device fails when and only when the second circuit
fails. Let X and Y be the times at which the first and second circuits fail, respectively. X
and Y have joint probability density function
(
6e−x e−2y for 0 < x < y < ∞,
f (x, y) =
0
otherwise.
What is the expected time at which the device fails?
(A) 0.33
(B) 0.50
(C) 0.67
(D) 0.83
(E) 1.50
18. [4-51]
Once a fire is reported to a fire insurance company, the company makes an initial estimate,
X, of the amount it will pay to the claimant for the fire loss. When the claim is finally
settled, the company pays an amount, Y , to the claimant. The company has determined
that X and Y have the joint density function
f (x, y) =
2
y −(2x−1)/(x−1) ,
x2 (x − 1)
6
x > 1, y > 1.
Math 370/408 Spring 2008
Actuarial Exam Practice Problem Set 4
Given that the initial claim estimated by the company is 2, determine the probability that
the final settlement amount is between 1 and 3.
(A) 1/9
(B) 2/9
(C) 1/3
(D) 2/3
(E) 8/9
19. [4-53]
A company is reviewing tornado damage claims under a farm insurance policy. Let X be
the portion of a claim representing damage to the house and let Y be the portion of the
same claim representing damage to the rest of the property. The joint density function of
X and Y is
(
6(1 − (x + y)) for x > 0, y > 0, x + y < 1,
f (x, y) =
0
otherwise.
Determine the probability that the portion of a claim representing damage to the house is
less than 0.2.
(A) 0.360
(B) 0.480
(C) 0.488
(D) 0.512
(E) 0.520
20. [4-54]
An auto insurance policy will pay for damage to both the policyholder’s car and the other
driver’s car in the event that the policyholder is responsible for an accident. The size of
the payment for damage to the policyholder’s car, X, has a marginal density function of
1 for 0 < x < 1. Given X = x, the size of the payment for damage to the other driver’s
car, Y , has conditional density of 1 for x < y < x + 1. If the policyholder is responsible
for an accident, what is the probability that the payment for damage to the other driver’s
car will be greater than 0.500?
(A) 3/8
(B) 1/2
(C) 3/4
(D) 7/8
(E) 15/16
21. [4-55]
An insurance policy is written to cover a loss X, where X has density function
(
3 2
x for 0 ≤ x ≤ 2,
f (x) = 8
0
otherwise.
The time (in hours) to process a claim of size x, where 0 ≤ x ≤ 2, is uniformly distributed
on the interval from x to 2x. Calculate the probability that a randomly chosen claim on
this policy is processed in three hours or more.
(A) 0.17
(B) 0.25
(C) 0.32
(D) 0.58
(E) 0.83
22. [4-56]
Two insurers provide bids on an insurance policy to a large company. The bids must be
between 2000 and 2200. The company decides to accept the lower bid if the two bids differ
by 20 or more. Otherwise, the company will consider the two bids further. Assume that
the two bids are independent and are both uniformly distributed on the interval from 2000
to 2200. Determine the probability that the company considers the two bids further.
(A) 0.10
(B) 0.19
(C) 0.20
23. [4-102]
7
(D) 0.41
(E) 0.60
Math 370/408 Spring 2008
Actuarial Exam Practice Problem Set 4
Let X and Y be continuous random variables with joint density function
(
8
xy for 0 ≤ x ≤ 1, x ≤ y ≤ 2x,
f (x, y) = 3
0
otherwise.
Calculate the covariance of X and Y .
(A) 0.04
(B) 0.25
(C) 0.67
(D) 0.80
(E) 1.24
24. [4-103]
A device runs until either of two components fails, at which point the device stops running.
The joint density function of the lifetimes of the two components, both measured in hours,
is
(
1
(x + y) for 0 < x < 3 and 0 < y < 3,
f (x, y) = 27
0
otherwise.
Calculate the probability that the device fails during its first hour of operation.
(A) 0.04
(B) 0.41
(C) 0.44
(D) 0.59
(E) 0.96
25. [4-104]
A family buys two policies from the same insurance company. Losses under the two policies
are independent and have continuous uniform distributions on the interval from 0 to 10.
One policy has a deductible of 1 and the other has a deductible of 2. The family experiences
exactly one loss under each policy. Calculate the probability that the total benefit paid to
the family does not exceed 5.
(A) 0.13
(B) 0.25
(C) 0.30
(D) 0.32
(E) 0.42
26. [4-106]
An insurance company sells two types of auto insurance policies: Basic and Deluxe. The
time until the next Basic Policy claim is an exponential random variable with mean two
days. The time until the next Deluxe Policy claim is an independent exponential random
variable with mean three days. What is the probability that the next claim will be a Deluxe
Policy claim?
(A) 0.172
(B) 0.223
(C) 0.400
(D) 0.487
(E) 0.500
27. [4-107]
An insurance company insures a large number of drivers. Let X be the random variable representing the company’s losses under collision insurance, and let Y represent the
company’s losses under liability insurance. X and Y have joint density function
(
1
(2x + 2 − y) for 0 < x < 1 and 0 < y < 2,
f (x, y) = 4
0
otherwise.
What is the probability that the total loss is at least 1?
(A) 0.33
(B) 0.38
(C) 0.41
8
(D) 0.71
(E) 0.75
Math 370/408 Spring 2008
Actuarial Exam Practice Problem Set 4
28. [4-108]
The stock prices of two companies at the end of any given year are modeled with random
variables X and Y that follow a distribution with joint density function
(
2x for 0 < x < 1, x < y < x + 1,
f (x, y) =
0
otherwise.
What is the conditional variance of Y given that X = x ?
(A) 1/12
(B) 7/6
(C) x + 1/2
(D) x2 − 1/6
(E) x2 + x + 1/3
29. [4-109]
Claim amounts for wind damage to insured homes are independent random variables with
common density function
(
3x−4 for x > 1,
f (x) =
0
otherwise.
where x is the amount of a claim in thousands. Suppose 3 such claims will be made. What
is the expected value of the largest of the three claims?
(A) 2025
(B) 2700
(C) 3232
(D) 3375
(E) 4500
30. [4-110]
A company insures homes in three cities, J, K, L. The losses occurring in these cities are
independent. The moment-generating functions for the loss distributions of the cities are
MJ (t) = (1 − 2t)−3 ,
MK (t) = (1 − 2t)−2.5 ,
ML (t) = (1 − 2t)−4.5 .
Let X represent the combined losses from the three cities. Calculate E(X 3 ).
(A) 1, 320
(B) 2, 082
(C) 5, 760
(D) 8, 000
(E) 10, 560
31. [4-111]
An insurance contract reimburses a family’s automobile accident losses up to a maximum
of two accidents per year. The joint probability distribution for the number of accidents of
a three person family (X, Y, Z) is p(x, y, z) = k(x + 2y + z), where x = 0, 1, y = 0, 1, 2, z =
0, 1, 2, and x, y, z are the numbers of accidents incurred by X, Y , Z, respectively. Determine
the expected number of unreimbursed accident losses given that X is not involved in any
accidents.
(A) 5/21
(B) 1/3
(C) 5/9
(D) 46/63
(E) 7/9
32. [4-112]
Suppose the remaining lifetimes of a husband and wife are independent and uniformly
distributed on the interval [0, 40]. An insurance company offers two products to married
couples: one which pays when the husband dies, and one which pays when both the husband
and wife have died. Calculate the covariance of the two payment times.
(A) 0.0
(B) 44.4
(C) 66.7
9
(D) 200.0
(E) 466.7
Math 370/408 Spring 2008
Actuarial Exam Practice Problem Set 4
33. [4-115]
Let X and Y denote the values of two stocks at the end of a five-year period. X is uniformly
distributed on the interval (0, 12). Given X = x, Y is uniformly distributed on the interval
(0, x). Determine Cov(X, Y ) according to this model.
(A) 0
(B) 4
(C) 6
(D) 12
(E) 24
34. [4-113]
A company offers earthquake insurance. Annual premiums are modeled by an exponential
random variable with mean 2. Annual claims are modeled by an exponential random
variable with mean 1. Premiums and claims are independent. Let X denote the ratio of
claims to premiums. What is the density function of X?
(A)
1
2x+1
(B)
(C) e−x
2
(2x+1)2
10
(D) 2e−2x
(E) xe−x